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Pixelating Relations and Functions Without Adding Substructures

Eldar Fischer

TL;DR

A generalization of a lemma from Ben-Eliezer, Fischer, Levi and Yoshida, ITCS 2021 is presented, showing that with a small amount of modification the authors can replace a hard-coded model with a pixelated one that has a finite description, in a way that preserves all universally quantified statements over the relations.

Abstract

We investigate models of relations over a bounded continuous segment of real numbers, along with the natural linear order over the reals being provided as a "hard-coded" relation. This paper presents a generalization of a lemma from [Ben-Eliezer, Fischer, Levi and Yoshida, ITCS 2021], showing that with a small amount of modification (measured in terms of the Lebesgue measure) we can replace such a model with a "pixelated" one that has a finite description, in a way that preserves all universally quantified statements over the relations, or in other words, without adding any new substructures.

Pixelating Relations and Functions Without Adding Substructures

TL;DR

A generalization of a lemma from Ben-Eliezer, Fischer, Levi and Yoshida, ITCS 2021 is presented, showing that with a small amount of modification the authors can replace a hard-coded model with a pixelated one that has a finite description, in a way that preserves all universally quantified statements over the relations.

Abstract

We investigate models of relations over a bounded continuous segment of real numbers, along with the natural linear order over the reals being provided as a "hard-coded" relation. This paper presents a generalization of a lemma from [Ben-Eliezer, Fischer, Levi and Yoshida, ITCS 2021], showing that with a small amount of modification (measured in terms of the Lebesgue measure) we can replace such a model with a "pixelated" one that has a finite description, in a way that preserves all universally quantified statements over the relations, or in other words, without adding any new substructures.
Paper Structure (7 sections, 10 theorems)

This paper contains 7 sections, 10 theorems.

Table of Contents

  1. Introduction
  2. The basics
  3. Presentation of the main result
  4. Homogeneous functions and inlays
  5. A Ramsey-type lemma
  6. Proof of the main result
  7. Discussion and variants

Key Result

Lemma 2.1

For every measurable set $A\subseteq \mathbb{I}^d$ and $\epsilon$ there exists $l$, so that for every $l'\geq l$ there is a set $B$ consisting of the (disjoint) union of sets of the type $\prod_{j=1}^d \mathbb{I}_{i_j,l'}$, so that $\lambda(B\Delta A)\leq\epsilon$.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Lemma 3.4
  • ...and 19 more